Hostname: page-component-8448b6f56d-c47g7 Total loading time: 0 Render date: 2024-04-23T17:03:01.981Z Has data issue: false hasContentIssue false
  • English
  • Français

Laplace transform identities for the volume of stopping sets based on Poisson point processes

Part of: Stochastic processes Geometric probability and stochastic geometry Stochastic analysis

Published online by Cambridge University Press:  21 March 2016

Nicolas Privault
Nicolas Privault*
Affiliation:
Nanyang Technological University
*
∗ Postal address: Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371. Email address: nprivault@ntu.edu.sg
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We derive Laplace transform identities for the volume content of random stopping sets based on Poisson point processes. Our results are based on anticipating Girsanov identities for Poisson point processes under a cyclic vanishing condition for a finite difference gradient. This approach does not require classical assumptions based on set-indexed martingales and the (partial) ordering of index sets. The examples treated focus on stopping sets in finite volume, and include the random missed volume of Poisson convex hulls.

Keywords

Poisson point processstopping setgamma-type distributionGirsanov identityanticipating stochastic calculus

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 60G57: Random measures 60G48: Generalizations of martingales 60H07: Stochastic calculus of variations and the Malliavin calculus
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 47 , Issue 4 , December 2015 , pp. 919 - 933
Copyright
Copyright © Applied Probability Trust 2015 

References

Abramowitz, M. and Stegun, I. A. (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Nat. Bureau Standards Appl. Math. Ser. 55), 9th edn. Dover, New York.Google Scholar
Baumstark, V. and Last, G. (2009). Gamma distributions for stationary Poisson flat processes. Adv. Appl. Prob. 41, 911–939.Google Scholar
Breton, J.-C. and Privault, N. (2014). Factorial moments of point processes. In Stoch. Process. Appl. 124, 3412–3428.Google Scholar
Cowan, R. (2006). A more comprehensive complementary theorem for the analysis of Poisson point processes. Adv. Appl. Prob. 38, 581–601.Google Scholar
Cowan, R., Quine, M. and Zuyev, S. (2003). Decomposition of gamma-distributed domains constructed from Poisson point processes. Adv. Appl. Prob. 35, 56–69.Google Scholar
Kurtz, T. G. (1980). The optional sampling theorem for martingales indexed by directed sets. Ann. Prob. 8, 675–681.Google Scholar
Miles, R. E. (1970). On the homogeneous planar Poisson point process. Math. Biosci. 6, 85–127.Google Scholar
Molchanov, I. (2005). Theory of Random Sets. Springer, London.Google Scholar
M⊘ller, J. and Zuyev, S. (1996). Gamma-type results and other related properties of Poisson processes. Adv. Appl. Prob. 28, 662–673.Google Scholar
Privault, N. (2012). Girsanov identities for Poisson measures under quasi-nilpotent transformations. Ann. Prob. 40, 1009–1040.Google Scholar
Privault, N. (2012). Moments of Poisson stochastic integrals with random integrands. Prob. Math. Statist. 32, 227–239.Google Scholar
Privault, N. (2016). Combinatorics of Poisson stochastic integrals with random integrands. To appear in Stochastic Analysis for Poisson Point Processes: Malliavin Calculus, Wiener–Itô Chaos Expansions and Stochastic Geometry . (Bocconi and Springer Series, 7), eds. Peccati, G. and Reitzner, M., Springer, Berlin.Google Scholar
Roman, S. (1984). The Umbral Calculus (Pure Appl. Math. 111), Academic Press, New York.Google Scholar
Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin.Google Scholar
Zuyev, S. (1999). Stopping sets: gamma-type results and hitting properties. Adv. Appl. Prob. 31, 355–366.Google Scholar
Zuyev, S. (2006). Strong Markov property of Poisson processes and Slivnyak formula. In Case Studies in Spatial Point Process Modeling (Lecture Notes Statist. 185), Springer, New York, pp. 77-84.Google Scholar